We have constructed a torsional resonance apparatus for contact‐free measurements of the electrical conductivity of solids. Owing to its use of modulation spectroscopy, the technique achieves considerable enhancement of sensitivity compared to previous contactless methods. Conductivities in the range 10−5mho/cm to 10−6mho/cm can now be measured by the contactless technique. Measurements of magnetic susceptibility as low as 10−12can be made. This sensitivity allows for investigation of magnetic impurities as dilute as 1 part in 107. In samples which display both phenomena of electrical conductivity and magnetic susceptibility, the two effects can be sorted out via analysis of frequency and/or temperature dependence. The present apparatus has potential for considerable improvement.